Non-convergence of some non-commuting double ergodic averages

Let S and T be measure-preserving transformations of a probability space (X,B,μ). Let f be a bounded measurable functions, and consider the integrals of the corresponding `double' ergodic averages: \[1/n∑ᵢ₌₀ⁿ⁻¹ ∫ f(S^ix)f(T^ix)\ dμ(x) (n≥ 1).\] We provide a new construction of diverse examples for which these integrals do not converge as n→∞. These include examples in which S and T are rigid, and hence have entropy zero, answering a question of Frantzikinakis and Host. We begin with a corresponding construction of pairs of unitary operators on a complex Hilbert space, and then construct transformations of a Gaussian measure space from them.